# Numerical algorithm for the space-time fractional Fokker-Planck system with two internal states

## TL;DR

This paper develops and analyzes a numerical scheme combining $L_1$ time discretization and finite element spatial approximation for a two-state fractional Fokker-Planck system with fractional Laplacian, providing error estimates and numerical validation.

## Contribution

It introduces a novel numerical method for the two-state fractional Fokker-Planck system with fractional Laplacian, including rigorous error analysis without regularity assumptions.

## Key findings

- The scheme achieves optimal error estimates in space and time.
- Numerical experiments confirm the theoretical convergence rates.
- The method effectively handles the nonlocal fractional Laplacian operator.

## Abstract

The fractional Fokker-Planck system with multiple internal states is derived in [Xu and Deng, Math. Model. Nat. Phenom., $\mathbf{13}$, 10 (2018)], where the space derivative is Laplace operator. If the jump length distribution of the particles is power law instead of Gaussian, the space derivative should be replaced with fractional Laplacian. This paper focuses on solving the two state Fokker-Planck system with fractional Laplacian. We first provide a priori estimate for this system under different regularity assumptions on the initial data. Then we use $L_1$ scheme to discretize the time fractional derivatives and finite element method to approximate the fractional Laplacian operators. Furthermore, we give the error estimates for the space semidiscrete and fully discrete schemes without any assumption on regularity of solutions. Finally, the effectiveness of the designed scheme is verified by numerical experiments.